Copyright K.Cuthbertson, D. Nitzsche 1 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture

Copyright K.Cuthbertson, D. Nitzsche 1

FINANCIAL ENGINEERING:DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001)

K. Cuthbertson and D. Nitzsche

Lecture

Value at Risk: Mapping

Version 1/9/2001


VaR for Different Assets (Mapping)

Stocks

Foreign Assets

Coupon Paying Bonds

Other Assets

(All of above have portfolio returns that are approximately ‘linear’ in individual returns ~ hence use VCV method)

Topics


Your Cash : Is it Safe in Their Hands ?

*

?


VaR for Different Assets (Mapping)


VaR for Different Assets: Practical Issues

PROBLEMS

STOCKS : Too many covariances [= n(n-1)/2 ]

FOREIGN ASSETS : Need VaR in “home currency”

BONDS: Many different coupons paid at different times

DERIVATIVES: Options payoffs can be highly non-linear (ie. NOT normally distributed)


VaR for Different Assets: Practical Issues

SOLUTIONS = “Mapping”

STOCKS : Within each country use “single index model” SIM

FOREIGN ASSETS : Treat asset in foreign country = “local currency risk”+ spot FX risk

BONDS: treat each bond as a series of “zeros”

OTHER ASSETS: Forward-FX, FRA’s Swaps: decompose into ‘constituent parts’.

DERIVATIVES: ~ next lecture


Stocks/Equities and SIM


“Mapping” Equities using SIM

Problem : Too many covariances to estimate

Soln. All n(n-1)/2 covariances “collapse or mapped” into m and the asset betas (n-of them)

Single Index Model:

Ri = ai + bi Rm + ei

Rk = ak + bk Rm + ek

assume Ei k = 0 and cov (Rm , e) = 0

All the systematic variation in Ri AND Rk is due to Rm

‘p’ = portfolio of stocks held in one country (Rm , m) for eg. S&P500 in US


Intuition

1) In a diversified portfolio ) =0

2) Then each return-i depends only on the market return (and beta). Hence ALL variances and covariances also only depend on these 2 factors.

We then only require( n-betas + m )to calculate ALL our

inputs for VaR.

3) = 1 because (in a well diversified portfolio) each return moves only with Rm

4) We end up with [1] VaRp = Vp p (1.65 m )or equivalently VaRp = (Z C Z’ )1/2 where C is the unit matrix


THE MATHS OF SIM AND VaR - OPTIONAL !

SIM implies: i,k =cov(Ri,Rk )= i k

m

i,2 = var(Ri) =

i m + )

BUT in diversified portfolio ) = 0 and = 1


We have(linear): Rp = w1R1 + w2 R2 + ….

From the SIM we can deduce that for a PORTFOLIO of equities in one country

Standard Deviation of the PORTFOLIO is given by :p = p m where: p = wi i

(ie. portfolio beta requires, only n-beta’s)

Hence: [1] VaRp = Vp p (1.65 m )

THE MATHS OF SIM AND VaR - OPTIONAL !


Alternative Representation

Eqn [1] above can be written:

VaRp = (Z C Z’ )1/2

where:

VaR1 = V1 1.65 ( 1 m ) and VaR2 =V2 1.65 ( 2 m)

Z = [ VaR1 , VaR2 ]

C is the identity matrix C = ( 1 1 ; 1 1 ), since = 1 for the SIM and a well diversified portfolio.


SIM: Checking the Formulae

[1] VaRp = Vp p (1.65 m ) = Vp ( wi i) (1.65 m)

= (V1 1 + V2 2 ) (1.65 m)

The above can easily be shown to be the same as

VaRp = (Z C Z’ )1/2

= (1.65 m) [ (V1 1) 2 + (V2 2)

2 + 2 V1 1 V2 2 ]1/2

where VaR1= V1 1.65( 1 m)

VaR2 =V2 1.65 ( 2 m )

and Z = [ VaR1 , VaR2 ] and C = ( 1 1 ; 1 1 )


“Mapping” Foreign Assets

into Domestic Currency


“Mapping” Foreign Assets into Domestic Currency

US based investor with DM140m in DAX

Two sources of risk

a) variance of the DAX

b) variance of $-DM exchange rate

c) one covariance/correlation coefficient (between FX-rate and DAX)

eg. Suppose when DAX falls then the DM also falls - ‘double whammy’ from this positive correlation

2s

2DAX



US based investor with DM140m in DAX

FX rate : S = 0.714 $/DM ( 1.4 DM/$ )

Dollar initial value Vo$ = 140/1.4 = $100m

Linear R$ = RDAX + R$/DM

above implies wi = Vi / V0$ = 1 and

p =

Dollar-VaRp = Vo$ 1.65 p

= correlation between return on DAX and FX rate

21

222sDAXsDAX



Alternative Representation

Dollar VaR

Let Z = [ V0,$ 1.65DAX , V0,$ 1.65S ]

= [ VaR1 , VaR2 ]

V0,$ = $100m for both entries in the Z-vector

Then

VaRp = (Z C Z’ )1/2


“Mapping”Coupon Paying Bonds


“Mapping”Coupon Paying Bonds

Coupons paid at t=5 and t=7

Treat each coupon as a separate zero coupon bond

P is linear in the ‘price’ of the zeros, V5 and V7

We require two variances of “prices” V5 and V7 and

covariance between these prices.

5(dV / V) = D (dy5)

77

55 )1(

100

)1(

100

yyP

75 VVP


Coupon Paying Bonds

Treat each coupon as a zero

Calculate PV of coupon = price of zero, V5 = 100 / (1+y5)5

VaR5 = V5 (1.65 5)

VaR7 = V7 (1.65 7)

VaR (both coupon payments)

=

= correlation: bond prices at t=5 and t=7

(approx 0.95 - 0.99 )

[ ] /VaR VaR VaRVaR52

72

5 71 22


5 6 7Actual Cash Flow

$ 100 m

5 7RM Cash Flow

Weights , 1- , are chosen to ensure weighted average volatility based on RM values of at 5 and 7 equals the interpolated volatility at “6”

Mapping on to “standard” RMetrics Vertices


“MAPPING” OTHER ASSETS

SWAP:

= LONG FIXED RATE BOND AND SHORT AN FRN

FRA (6m x 12m)

= BORROW AT 6 MONTHS SPOT RATE +LEND AT 12M SPOT RATE

FORWARD FX:

= BORROW AND LEND IN DOMESTIC AND FOREIGN SPOT INTEREST RATES AND CONVERT PROCEEDS AT CURRENT SPOT-FX


End of Slides

Documents

Copyright K.Cuthbertson, D. Nitzsche 1 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture